Alex Bellos: “Alex Through The Looking Glass” “The Grapes of Math” | Talks at Google

Alex Bellos: “Alex Through The Looking Glass” “The Grapes of Math” | Talks at Google


PETER: Welcome to
Authors at Google and welcome to Alex Bellos. I don’t know if any of
you saw his earlier talk a couple of years ago around
his first book, “Adventures in Numberland,”
which was a huge hit. And on a personal business,
it was a huge hit for me because my 10-year-old
son at the time I couldn’t get him to
read any books at all. And in desperation,
we were on holiday, and I kind of handed him
“Adventures in Numberland” and said have a look at this. And he quickly became
extremely hooked on and spent the entire holiday
going yam, tan, tethera which I’m sure you’ll know is the
ancient English way of counting sheep. And when you get to
15, it’s bumfit I think I’m right in saying. And he spent the entire
holiday talking about bumfit. So it sort of both hooked
him in an interest in maths and gave him a license
to use bad language so it was kind of perfect. So the new book is “Alex
Through the Looking-glass.” Alex is going to
talk for half an hour or so and then take
some questions? ALEX BELLOS: A little bit
more than that but, because I want to be very informal,
so if there’s anything you don’t understand or you
want more, just shout out, put your hand up. So let’s kind of have
the kind of Q and A as we go I think because
we’re a little bit delayed. I’m going to have to– PETER: And I’ll hand
if over [INAUDIBLE]. Keep going. Cheers. ALEX BELLOS: Usually,
I’m used to giving talks to book festivals where the
kind of mathematical numeracy is going to be a lot lower
than you guys are here so I might whip through
it but you might have slightly more
high-brow questions. So thank you very much, Peter. I’m Alex Bellos. This is me. Oh, my god. I just pressed something. That’s me with a weird
stain on my red jumper. I don’t know why
I was given that. And this is my new book, “Alex
Through the Looking-glass,” which sounds very much like it’s
a sequel to “Alex’s Adventures in Numberland” but
it isn’t really. Both can be read separately. You don’t need to have read
the first one to read this. If there is a
difference, the first one was just an exploration about
mathematical abstraction and what we get when we use it. This is a little bit
more kind of applied. It’s about using
abstraction to learn about the world, mathematics
being the language that we use to talk about the
world, and also how life reflects numbers
and numbers reflect life. I also wanted to get into
kind of psychology of numbers and interesting
psychological responses that we have to numbers which
are quite surprising I think and definitely under-researched. Now, how do you write a
book on maths and numbers and make it interesting. It’s quite difficult because
the nature of the subject is dry and sometimes it’s
conceptually challenging. So I’ve got a degree
in maths and philosophy and then I became a journalist. So my approach is
always to tell stories. And I think that
by telling stories, if you can interest
someone in the story, then you don’t need
to say, oh, by the way I’m going to teach
you some maths. You just learn the
maths on the way. And where do you
look for stories? Well, the kind of classic
place to look for a story is in oneself. And so, actually, my story as
a popularizer of mathematics is really where
this book begins. And I have to say I don’t
think it happened at my talk at Google last time, but pretty
much every talk that I did when the first book came out,
I would give what I thought was a fantastically fascinating
talk about maths and numbers. And someone would put
their hand at the end of it and they would say, yes, sir. And they would just say, so
what’s your favorite number? And I can just
remember thinking, oh, have you learned nothing? Who has a favorite number? I’m grown-up. You’re a grown-up. And I would just think they
were kind of teasing me. It was way of belittling
the subject on myself. Until I kept on being
asked this question. I would give talks
at universities, secondary schools,
books festivals. People were always asking it. So once I just said,
oh, I didn’t know. What’s your favorite number? But rather than saying,
ah, they went, oh, 12, gave me a considered response. And I just was like, what? And then the person next
to them said, oh, no, my favorite number is 13. I said, well, who here’s
got favorite numbers? And at least half the
audience of grown-ups put their hands up. And this happened
a few more times. So I thought, well,
this is interesting. From being something that I
thought was completely idiotic, I thought, well, if
grown-up people who are literate and numerate
have serious [INAUDIBLE] emotional reactions
numbers, this is interesting and why don’t I try
and research it? And so what I did is I thought
I would undergo a survey to try and find the
world’s favorite number. But essentially it was
to try and quantify emotional responses to numbers. So I set up this website
here, favouritenumber.net. I realized it was going to
have to be international, so I also bought this
one, favoritenumber.net, just so no one was excluded. And within a few
weeks– at that time, four years ago, I maybe had
only a few thousand followers on Twitter. I just put it on Twitter. What’s your favorite number? Join my survey. Within a few weeks,
more than 30,000 people from around the
world had replied. And so this was already
vindicating this idea. This is something
that serious people feel passionately about it. Now, what were the results? The results were this. 7, 3, 8, 4, 5, 13, 9, 6, 2, 11. Which “Metro” said was
the most confusing top 10 list ever written. Which it is. Now, if you go back
to favoritenumber.net, I have all the results up there. There’s a big Excel file of all. There’s more than 1,000
individual numbers and all their frequencies. And you can get
them there if you are interested to do
an analysis on them, if you have a statistical mind. But what was interesting
was that when I launched the results, it
was an actual news story. Again, showing that people
are interested in this stuff. It was a news story
in the “Times.” It was a news story in the
“Daily Mail” of all places. But it also started
going around the world. This is “Glamour”
magazine in Paris. And I don’t think I
found one country that didn’t report it in some way. Just going to mention
a few of them. Hungary. Hungary has a huge brilliant
mathematical tradition. And per capita, I think
that more Hungarians entered the survey than
any other nation, so they were really
fascinated by it. Obviously, it was
[INAUDIBLE] Greek, Greece being the beginnings
of Western thought. By the time, it got
to India, I was now a Brazilian mathematician. But India– there
are certain countries where mathematics and the
idea of doing arithmetic is really an important
part of national identity. And so it was all over the
regional press in India. Vietnam, and obviously
I can’t read Vietnamese. And it just became
quite interesting what happens in the different
photo archives countries when you put in seven. This is what you get
when you’re in Vietnam. And when you’re in China,
this is what you get. So it was a new story. But what is the
moment that you really know that it has touched
the public consciousness, it’s become a real news story? It’s when you’re a
question– volume, please? Three days afterwards, it was
on “Have I Got News for You.” [VIDEO PLAYBACK] -Alex Bellos asked 44,000
people to submit their favorite number– ALEX BELLOS: So yeah there
were 44,000 people in the end. But I only did the cleaning
of the data for 33,000. The Nigel [INAUDIBLE] numbers. —top ten world’s
favorite numbers. -Yes. -Yeah. ALEX BELLOS: The
reason why this is funny is because we actually
do have feeling– things are only fun if there’s an element
of something serious about it. -Number ten, the tenth most
popular is, in fact, 11. ALEX BELLOS: [INAUDIBLE]
mathematical joke. -14. -No. -21? -No. At number nine is
the number two. The most popular number
in eighth position is six. And seven, the seventh most
popular number is nine. And the sixth most
popular number is 13. OK. Here’s the top five. -This is going to be on Channel
Four for a whole evening. There’ll be some talking
heads in a minute. What did you of six? Eh, not it. ALEX BELLOS: This is
actually my idea– -I’ve always liked five because
it’s a working class number. That’s what I like about five. But I think that
three [INAUDIBLE] and I thought three was a
magnificent num– oh, I’m sorry. I’ll stop auditioning. Sorry. Where am I? -Let’s firstly
complete the top five. The number five is five. And number four is four. And number three is eight. And number two is
three, the magic number. And, of course, seven is
the top one, number one. [END VIDEO PLAYBACK] ALEX BELLOS: OK. It’s interesting that
everyone laughed at five being five and four being
four because, essentially, we like small numbers. And the bigger the numbers
get, we like them least. This is what the survey says. So a number should, if there
are no other considerations punch more or less exactly
their numerical value. So those are almost the ones
that are the least funny or the least interesting. And what happens
and what you see is the odd numbers and
especially prime numbers are way up the rankings
and round numbers really, really way back
down the rankings. We do not really
like round numbers, but we really like
prime numbers. And that is
something which, even though the quantitative research
you can dismiss and say, oh, this is silly because
it’s 44,000 people but it was 44,000 people who
already feel passionately about it who selected
themselves to join the survey. Even amongst them, there
are very coherent patterns of these emotional responses. So on the survey, I said,
what’s your favorite number? But I also said, why? I just left it blank. There’s a little space. You could write it in there. And to me, it was the
qualitative results of the survey that were
actually a bit more interesting and a bit more
surprising and that felt a bit like
proper new research. So I’m going to just take you
through some of the entries with some of the reasons. And we start to
understand what it is about numbers that make
people excited by them. So this number was
selected, 10 over root two. Now, in this
audience, does anyone know what 10 over root
two is in decimals? It’s 10 times 1 over root two. Which if you work out is
actually square root of 50. 7.07 which in America is
called the Boeing number because of 707. And I had originally thought
that Boeing was called 707 just because this looks a
little bit like a plane, two wings and a fuselage. Apparently that’s
not the reason why. But you will find that certain
numbers work in brands really well based on just how they
look and how they sound. Because even though
numbers are supposedly these abstract entities
just representing quantity and order. How do we get to them? They exist within
culture, within language. They have to look like
something as a symbol. And all these things
actually influence how we understand even
the numerical properties of numbers. I’m going to read you the
reason why the person chose 707. It wasn’t because they were
an aviation enthusiast. People wrote down their
nationalities, their genders, their level of mathematical
ability, and their age. And this was from a
woman in Canada, aged 34, with university-level maths. She said, this number
is 10 root two. I’d been doing a
lot of trig homework for a calculus class I
was taking in undergrad and this number appears a lot. At the time, I was sort
of weirded out by the fact that I kept waking up at 7:07
in the morning instead of 7:30 when my alarm was set for. Anyway, one Saturday, I went
to my local art supply store and bought some paintbrushes. To my surprise, the
total came to $7.07. And I sort of blurted out,
oh, that’s 10 over root two all over again. So the very cute
cashier, for whom I had a rather pathetic crush
and who I was constantly embarrassing in front of,
after explaining myself, he was duly impressed and
began embarrassing himself in front of me whenever
I came to the store. And from that point
on, I realized there’s a brand of arty guys
that like nerdy girls and this still makes me happy
some 15 years later. OK. This is written in a tiny
little white space on a website. Just the story, the passion. There’s so much stuff
that you could analyze. Also the fact that when you
start to think emotionally about a number you
just see it absolutely everywhere even though numbers
don’t really predominant one over the other in
such a strong way. 1,000,000,007. Why was this chosen? This was chosen by
21-year-old male Russian because it’s the largest
prime number I can remember. So the fact that people take to
remembering large prime number. A prime number I’m sure you now. A prime number is a number that
only divides by itself and one. So they start with two. Two is the only even
prime number, then it goes 3, 5, 7, 11,
13, 17, 19, et cetera. Now, 1,000,000,007
is interesting because 1,000,000,009
is also a prime number. In fact, you get this things
called twin primes which are prime numbers
that are two apart. And it’s conjectured that there
are an infinite number of them. But the largest one
that’s know of the form one, loads of zeroes, then
seven, one, loads of zeroes, then nine is 1,000,000,007
and 1,000,000,009. So you can kind of
think you’re really clever to remember
[INAUDIBLE] prime number. It would be clever if it
was the second highest prime number I can remember. 219. Now, this was chosen by
an 18-year-old Irishman because it’s the lowest
whole number that does not have its own Wikipedia entry. So it’s too boring. It’s the lowest boring
number, basically. Every number from one to 218
has it’s own Wikipedia entry. And, actually, this was
a couple years ago now. So I’ve looked. There’s an inflation in
lowest boring numbers. And now the lowest
boring number is 224. OK. But you might not
think that Wikipedia is a very good judge of what
makes a number interesting or not. And so I thought I would look
in the online encyclopedia of integer sequences, which is
a kind of Bible of mathematics and really of all science. It contains about 250,000
separate sequences. And a sequence is just
a list of numbers. They can be infinite. They can be finite. So the prime numbers,
that’s a sequence. The square numbers, at
one, four, nine, sixteen, twenty-five. That’s a sequence. What it be interesting
to find out what is the lowest number
that has never appeared in the online encyclopedia
of digit sequences, which you could probably say is the
only number– well, probably the lowest number never
to have to appeared in any mathematical
piece of research in the last 3,000 years
since, really, there’s been mathematics. And that number
is 14,228, which I thought was quite
low personally. But it’s also interesting. It’s very, very
divisible, isn’t it? Divided by two obviously. This idea that numbers
that are really interesting are the prime numbers,
the odd numbers. Finally, this number 11,
which is also a cheap plug because I also
used to live in Brazil and wrote a book on Brazilian
football, which has also been updated for the World Cup. So if you’re interested
in finding out about Brazil and the World
Cup buy my book about it. It has [INAUDIBLE], who
is the– well, he’s not the captain, actually,
at the moment, but he’s the talisman of
the Brazilian national team. And the reason why 11 was
chosen was from an American, 36 years old, and I like to
think that he was from the deep South because if he wasn’t, it
doesn’t really make much sense because he said, I like 11
because it sounds like lovin’. So, why? What are the reasons
why seven was the world’s favorite number? I made a little animation,
hoping the sound is OK. [VIDEO PLAYBACK] -We’ve been obsessed by the
number seven for as long as we know. Go back to the
earliest writings there is, on Babylonian clay
tablets, and they’re just full of sevens. Then you have seven dwarves,
seven sins, seven seas, seven sisters. The list just goes on and on. So why is seven so special? One argument is that
there are seven planets in the sky visible
to the naked eye. To me, that’s just coincidence. There’s a much more
compelling reason. Seven is the only number
among those that we can count on our hands– that’s
those from one to 10– that cannot be divided or
multiplied within the group. So one, two, three, four, and
five, you can double them. six, eight and 10,
you can half them. And nine you can
divide by three. Seven is the only
one that remains. It’s unique. It’s a loner, the outsider. And humans interpret this
arithmetical property in cultural ways. By associating seven
with a group of things, you kind of makes
them special too. The point here is
that we’re always sensitive to
arithmetical patterns and this influences our
behavior even if we’re not conscious of it and irrespective
of our ability at maths. [END VIDEO PLAYBACK] ALEX BELLOS: There’s
another thing which kind of ties into this. If you were to ask someone– and
this is a psychology experiment that’s been repeated many
times– just think of a number off the top of your
head between one and 10, most people say seven. And why they say seven,
essentially– again, you’re doing mathematics
without realizing. You think, well, I’m
not going to say one because that’s not off
the top of my head. That’s just too obvious. It’s not sort of
arbitrary enough. I’m not going to say 10. I’m not going to say five. Well, you’ve just done
the two times table. And then you sort of say,
well, two, that’s also. I wouldn’t choose two. Then you eliminate
two, four, six, eight. You do exactly what
I was doing in there and you’re left with seven. Seven– it’s the most difficult. It’s the one that’s
[INAUDIBLE] to stretch where it feels the most random. Now, if we go back to
the qualitative results of the favorite number survey,
I was trying to work out, is there any way that I
could grasp or explain the kind of collective
views about numbers? And I was thinking,
well, how can I do this. And I was reading all this
more than 30,000 reasons why people like numbers. And I noticed that the language
used to describe each number was different. So I’m going to
pluck the adjectives that we use to describe
different numbers. Number one was
described– this is each word in a
different reason– as independent, strong,
honest, brave, straightforward, pioneering, competitive,
dramatic, egoistic, hot-headed, and lonely. So imagine we were a
creative writing class. And I said this is like the
main protagonist of our story. Those are really quite coherent
personality traits there. Two. These are words used to
describe the number two. In fact, sorry. That was number one. Two. So look at number two and see
if you visualize these traits. Simple, elegant, cautious, wise,
observant, pretty, fragile, open, sympathetic,
comfortable, flexible, subtle, inconspicuous. This is from lots
of different people and they’ve got lots
of different reasons but together, again, there
is a kind of coherence there. And what struck me most of
all was that if these were a creative writing
class and we were having to imagine characters for
the numbers one and two, one is really male
characteristics or traditionally
male characteristics. And two, traditionally
female characteristics. So the cartoonist of the
book did exactly that. This is how she interpreted it. We could do a fantastic kind
of rom-com or romantic story with one as the man
and two as the woman. Now, I saw this and
I thought, well, this is really quite striking. But the more that I read–
and I read quite a lot of stuff on the history
of maths and the growth of intellectual ideas and
also about psychology– it’s really quite
common, [INAUDIBLE]. So numbers were invented
about 8.000 years ago, say, in Sumeria, which
is present-day Iraq. The very first
words that we know that we used by humans for
the number one and the number two– the word
for the number one is the same words as
the word for phallus and the word for two was
the same word for woman. So right from the beginning,
there– for whatever reasons. We don’t know. You can imagine, I suppose. One is male and two is female. Actually, a friend of mine who’s
a quite well known novelist, she said well, Alex, I
was reading your stuff about masculine and feminine. And I just thought that
everyone thought that. I’ve always thought– and she’s
one of the smartest people that I know. And she was saying, yeah,
well because it’s obvious that one and odd numbers
are masculine and two and even numbers are feminine. They obviously are because two
you can kind of split apart. They can become two new things. It’s like they
kind of give birth. And obviously, I had
never thought that at all. But the fact that
someone– there are people who are
smart who think these. We do relate to numbers
in emotional ways. Pythagoras who’s the
beginning of Greek mathematics had this Pythagorean
brotherhood. And they believed that
odd numbers are masculine and even numbers are feminine. And I discovered
some research that had been done quite recently
in which respondents or participants were asked
to look at babies who are six weeks old of
indeterminate gender and all you have to do is
just say, is this a boy or is this a girl. Always, they had numbers
at the bottom of them. But the people were told. Don’t look at the numbers. There’s going to be a number. But just don’t look at it,
just look at the child. And what the
experimenters were doing is that sometimes it was all
three even numbers, sometimes all three all numbers,
all kind of randomized, proper science and that. And it turns out that,
with odd numbers, you’re 10% more likely
to think that a baby of indeterminate gender is male
and with even numbers, 10% more likely that the baby is female. So there is still this kind
of subconscious association with odd numbers with
masculinity and even numbers with femininity. Also, when I [INAUDIBLE] started
to get interested in this idea that numbers have personalities. And even though if I had
been talking to myself a few years ago I
would have thought that absolutely ridiculous. When I started to
think about it, I though, yeah, well I
use numbers all the time. Yeah, certain numbers do kind of
feel a bit different to others. And then when you
start looking at even Shakespeare noticed this. This is from “Merry
Wives of Windsor,” Falstaff, They say there
is divinity in odd numbers, either in nativity,
chance or death. And it’s true. All our most mystical
religious symbolic numbers are odd numbers in
the Western tradition. So three, Christianity,
seven, 13, five a little bit. They’re all low primes
and this is interesting. And there is
psychological research, proper psychological
research, not just my online survey,
that starts to give us a bit of an inkling about
why this might be the case. So one of the experiments was
you’re just shown two digits. Either they’re both odd
or they’re both even or one is even and one is odd. All you need to do is
just press a button when they’re both the same. So either when they’re both
even or when they’re both odd. Turns out you’re much slower
pressing the button when they’re both odd and
you make more mistakes. In other words, it kind of takes
us longer just to process it, just to think about them,
not to do any math with them, just to think about odd numbers
than it does with even numbers. And this sort of gives this
idea that somehow there’s sort of more room,
there’s more space, to have a kind of emotional
interaction with them. The final experiment I
want to talk about in terms of these psychological and
emotional responses to numbers is all about branding. And I’m sure we’re going to
be seeing a lot more of this because this is
quite recent work. The scientists, the
academics, wanted to know if we are more
likely to be attracted to want to buy and pay
more money for brands if they have different
types of numbers on them. So WD-40 is a very
popular brand. That’s an even number. Oxy10. That’s an even number. There’s something sensible,
safe about even numbers. And it turns out that, for
household products, for things that you want to
work and be reliable, people are much more
likely, will pay more money, if they have an
even number on them. So the experiment here was
two packs of contact lens, one is the brand called
Solus36, one is Solus37. They’re absolutely
exactly the same. You do all these tests about
how to evaluate the brand. Solus36 is much more popular. People pay more money for it. They want it. They like it better. OK. So this is the first
of the experiment. Then they add in this
tagline, 6 colors. 6 fits. Just to see if this affected how
much you wanted the brand how much you liked it,
how much money you were prepared to pay for it. And what happened
is that Solus36 becomes even more liked
and Solus37, which wasn’t liked very much
in the first place, becomes even less liked. And the argument put
forward, the hypothesis, by the scientists is that the
reason why we like Solus36 is that we– how do
we learn numbers? We learn numbers by
learning times tables by rote when we’re children. So we are much more
familiar with numbers that are in times tables. So 36. Whenever we do the six
times tables, we get it. We do the 12 times
tables, we get to it. And what happens? We never get to 37 because
it’s a prime number. It’s not on any times table. It will be the answer to very
few– basically, as a child, you will never utter or
think about the word 37 but you will use
36 all the time. So this idea that it’s
what you’re familiar with is that fluency of processing. And you misattribute this
fluency of processing for liking of the product. So it kind of feels good. Oh, I like that. Oh, it’s like a friend,
a friendly product. And this is what’s
exacerbated by the tagline because subconsciously
we’re seeing six, six, 36, and it’s like, oh, yeah. It’s like a nursery
rhyme from school. I remember that. I like that. I really like those
contact lenses. And with six, six, 37, there’s a
kind of mathematical cacophony. We don’t like it. So that brand is
really unattractive. And they did a lot
more experiments. And it’s interesting that when
you have an ad for something and if you subtly put numbers
in it that relate arithmetically together, you can
increase quite drastically the liking for that brand. So in the future
I’m sure there’ll be lots of numbers subtly
positioned within ads. Which you tend to
think, oh, god, I hated learning maths by rote. I hated learning
my times tables, but actually that either can
be kind of leveraged to make you feel good about a product
is ironic to say the least. That is essentially what is
the beginning of my book, the story about me and
the favorite numbers and emotional
connections to numbers which is how life
reflects numbers. The rest of the book,
so the majority of it, is kind of proper
more serious maths. And I’m going to just talk to
you about one part of it now that begins with– we’re
talking about prime number. So let’s just repeat
prime numbers. These are the prime numbers. The numbers that only divide
by themselves and one. Those are the first seven–
I think, yeah. –of them two, three, five, seven,
eleven, thirteen, seventeen. We live in the age of algorithms
as we’re told every day. What was probably the
very first algorithm in mathematics and,
therefore, in civilization? It was probably what’s called
the sieve of Eratosthenes which was the algorithm
to try and find all the prime numbers
under a certain number. So this is a grid of the
numbers from one to 100. I’ve done them in six rows. We’re going to use the
sieve of Eratosthenes to try and find all
the prime numbers. OK. What do you– you start at the
bottom, you get to a number, if the number has
not been eliminated, it’s a prime number. And then you eliminate
all multiples of it. So let’s start with one. One’s not a prime number
because we start at two. Get to two. Then we eliminate
all multiples of it which is basically
all even numbers. And it’s nice when you
do it in rows of six. So next one. We get to three. Three’s a prime number. Then we eliminate all
multiples of three which is just one more line
because the bottom row is all multiples of three but
we’ve already eliminated that. Four’s been eliminated. We get to five. That’s a prime number. Eliminate all multiples of that. Six has been eliminated. You get to seven, eliminate
all multiples of that. Nice little crisscross pattern. Then up to 11. And you only need to–
when doing the sieve, go to the square
root of the number that you’re seiving which
is root of 100 which is 10. So we’ve got to 10
already so therefore we can guarantee that all of those
numbers in their are primes. And there are few
interesting things to say. First is that all prime numbers
are either one or one less or one more than
a multiple of six because that can only be in
line one or in line five. If this went on to infinity,
it would be the same. But also– which is why
people are kind of fascinated by prime numbers– is that
before you get to them, there’s no way of knowing if
a number is going to be prim or not. They appear to be
kind of randomly sprinkled around
the number line. And that is one of the
reasons why they are still one of the first things
to be studied in numbers and there’s still–
lots of things about them are just not known. Lots of books about
prime numbers. So in 1963, Stanislav
Ulam, who was one of the great mathematicians
of the 20th century, was doodling, got
bored in a math lecture and started to doodle. And he did a great but he
put the one in the middle and then started to
spiral and then started to cross off the prime numbers. And he noticed
something that no one had noticed before–
that prime numbers tend to sit on lines together. And so this is something
after several thousand years no one had ever
bothered to do this. And this is up to 100,
which is the numbers that we saw on the grid before. But if we do it up to 20,000,
you see it a bit more. There are these lines that
prime numbers seem to sit on. And it’s called an Ulam spiral. And really, in maths, you
kind of have total order and that’s a bit boring. You have total chaos. That’s a bit complicated. When you have this sort of in
the bit between order and chaos which you have here is where
things get really interesting. So this is a really kind of
simple yet incredibly deep image of where are
all these primes and what relations do
they have to each other? This is Ulam right
here on the right. And he’s sitting. This is just after
the Second World War near Los Alamos with John
Von Neumann on the left, who I hope that Google has a
shrine to John Von Neumann somewhere because basically
he invented the computer with Alan Turing of course. But a lot of the kind
of conceptual framework of computing was basically
invented by John Von Neumann. In the middle is
Richard Feynmann, the famous American physicist. This is an amazing picture
because you could quite easily argue that these three
men sort of changed the 20th sort of
more than anyone else because they were
instrumental in creating nuclear weapons, the computer,
Von Neumann– possibly the greatest mathematical
genius of the 20th century. Whatever he tried to do he
invented like a new field. And it’s one of the fields
that he and Ulam invented I’m just going to finish
off talking about now which hopefully is right up your
street because it talks about essentially the first computer
craze that there ever was. So Von Neumann was
inventing so much stuff. And it was in the
1940s, early ’50s, started to worry about
what he was creating. Computers, robots, what’s
going to happen to robots? What would it take, he asked,
for a robot or a machine to replicate itself? And this is actually a kind
of psychologic problems. It’s like a
mathematical problem. It’s not really a
biological problem. It’s a kind of conceptual
idea because just say you’ve got the
machine and you’ve got, say, the instructions. If you put the instructions
in the machine, can the machine make itself
from the instructions? Because the
instructions, obviously, are part of the machines. So you cannot replicate. And basically it can’t
in a finite world. Just say you’ve got a machine
with a set of instructions. What you want to do is
create exactly that. A machine and a set
of instructions. Just say the machine
reads the instructions and it’s got instructions
how to rebuild a machine. Does that set of instructions
have instructions on how to build a new
set of instructions because if it does, then
that set of instructions within the instructions. It’s got to have some
other instructions. And then you get this
infinite regress. It’s a finite system. It can’t happen. So he realized that
if you have a machine with a set of instructions
to self-replicate, you need to treat the
instructions in two ways. You need to read it
to build the machine and then you need to
duplicate it and then present it to the new machine. So there are two
ways– basically, you need these two elements. And when, a few years
ago, Watson and Crick discovered DNA, they realized
that essentially the same thing happens in humans
more in general terms. The DNA is the instructions. It has instructions
for the cell. But DNA does not contain
instructions for DNA because DNA replicates. It’s a double helix and one
of the helixes splits off and then you create another one. So this is another fantastic
example of something which the mathematics was
conceptually discovered years before it was shown
to be that was the way– it’s like
the Higgs boson. It was kind of
mathematically proved and then, decades later,
it was discovered. What John Von Neumann
wanted to do– he’s also like a fantastic
entertaining character. He loved parties and
sometimes would hold parties just because he liked working
to the sound of parties. So he would be there
kind of mixing together and then go back to his room
and do some amazing maths. He wanted to think
about building a self-replicating
robot, something which can use instructions,
then duplicate them. But it was just too complicated
actually working out how to build it
in the real world so Ulam who, as we’ve seen,
loves grids and things like that and
patterns, said, let’s invent a new kind of
mathematical abstraction. Which they did, called
the cellular automaton. And a cellular automaton
essentially is a grid of cells. The cells can have
different states but the behavior of
the cells depends only on neighboring cells. I’m going to explain now a
very simple cellular automaton called the Game of Life
because Von Neumann invented it to try and find something
that would self-replicate and he managed to do
that conceptually. And then, a decade or
so later, at the end of the ’60s, a British
mathematician called John Conway, who’s
still alive– he’s at Princeton at the
moment– invented what is one of the most famous
examples of what’s called recreational maths but
it’s also quite serious called the Game of Life. It’s not a game. It’s a cellular automaton. Life because it emulates
evolution of life. So this is the grid. The game of Life– I’m going
to explain how it works. All of it takes place on
a two-dimensional grid. The grid can be infinite, and
it has these orthogonal cells. The cells can have two
states, either alive or dead. This is a live cell. All around it are dead. And it’s immediate
neighbors– there are eight of them– are these
ones here all around it. So its behavior depends only
on those eight cells around it. And these are the
rules that it obeys. There are the Game
of Life genetic laws. One, if you have a live, if
it’s surrounded by zero or one live neighbors, it
dies by loneliness. If it has two or three live
neighbors, it survives. Four or more live neighbors,
it dies by overcrowding. Really, really simple. Basically, it a live cell is
surrounded by two or three, it survives. If it’s not, it dies. A dead cell, if it has
exactly three live neighbors, it becomes alive. And in all other
cases, it stays dead. So Conway– this
was the late ’60s– he had one of those grids. It was basically a Go board. There were no computers then. And he used Go counters just
to see what would happen. So let’s see what happens. This is a pattern
with three shapes. What the Game of Life does
is what we will always do– we will start
with a pattern. And the idea is to put
an interesting pattern on the board and then you
don’t, in fact, touch it at all. It then just
evolves and changes. So what we want to
see is how it evolves. And Conway chose
his rules to make the most unpredictable
and interesting way that things evolve. So what we’re going to do here–
let’s look at this one here, surrounded by one live
neighbor, it will die. Surrounded by two live
neighbors, it will survive. Surrounded by one live
neighbor, it will die. A dead cell surrounded by
three, it will survive. So what happens is
that you work out how each one is going
to behave and then you apply the rules on
everything and then you get a new generation. So what would happen here? The new generation
would be that. OK. And then it dies because
there are only two left. This one here. That becomes that. And then it dies. This one here. We’re going with shapes
with three live cells. Becomes that. And then it just stays there. It’s called a stable form. So at the beginning, they
were like zoologists. They’ve got this new world. They were just
trying to work out what was there, finding the
patterns and what would happen. This one here, another
three cell pattern. Becomes that, then
that, and then that. That’s known as the blinker. OK. Not that interesting yet. When you start to add more,
most things end up dying, but sometimes you get
amazing bursts of life. So this four-cell pattern
becomes this then this then this then this. It’s four blinkers, which
is called traffic lights. Let’s go and look at
a five cell pattern. So they had done four-cell,
see what happened to them. They were cataloging them. And then they noticed
something amazing. And this is when basically
the study of life really took off because look
at this five-star pattern. It repeats. It’s called a glider. Every four generations,
it’s moved one space down and one cell along on the grid. So essentially it moves. It looks like it’s moving. And this was the– Conway
called these spaceships in the kind of etymology of the
kind of animals or the living things that you get. And he wondered and
in Scientific American he said, does anyone
know if there’s any pattern that can grow and
have more and more live cells. Because this one it all stays
at five cells all the way down. Something that can grow
and get infinitely big. And he put a reward in
Scientific American. And the Martin Gardener
article that wrote about it was the most read
article he ever wrote. And this is a guy
who is like the guru of all kind of popularization
of maths and science. And this was Conway in
Cambridge writing about it. Martin Gardener in
America writing about it. But the real development
started to happen at MIT. The Game of Life
came along at a time where computing
was just beginning. But you would only
have computers if you were in a big
kind of organization or if you were at a
university like MIT. And I don’t know
if you know where the word “hacking” comes from. The word “hacking” originally
comes from the MIT Model Railway Club, and a hack is when
you kind of customize something just for the hell of
it, just to make it fun. And so the hackers
were the people who worked at the MIT
Model Railway Club. But when computers
came along, they became not so interested
in model railways and worked on computers. And the thing with
Game of Life, as you saw– incredibly simple
rules but incredibly complex behavior. It became basically the
first computer craze that the hackers
at MIT– and this is essentially
the people whose– the forefathers or forebrothers,
say, of Steve Jobs, Bill Gates, et cetera, were
the hackers at MIT working on the Game of Life, which
was basically the biggest deal in computing
in the early ’70s. And Bill Gosper, who was the
kind of king of the hackers, discovered and won the prize
for the first pattern that grows without bound,
which is this. And it looks like a
kind of pair of lungs. And what it’s doing is
called a glider gun. And basically it just, every
period, spits out a glider. And that will carry
on, getting bigger. And the total of live cells
just gets bigger and bigger. And the more that
people looked at this, they started to find
other patterns that did other interesting things. So this thing here
is called an eater. And what an eater
does is an eater will eat anything
that you send at it. So this gave the
first indication that this might actually
have some use because it’s like how to build
self-repairing object. But also it meant that
you could start to just have these amazing kind
of engineering patterns where you could build
massive patterns and all the debris you could
have eater just clean them up. So let’s have a look at
the eater, what happens. So this is a glider coming in
it to an eater, gets eaten. This is another spaceship,
just gets eaten. And let’s now look at–
this is the sort of thing that people started to build. So this is a glider moving. And I’m going to
be going out of it. What’s happening? So it’s going to this
other massive pattern. What’s going to
happen to the glider? Bing. It bounces. So that’s basically
a glider reflector. It will bounce a
glider at 90 degrees. So as we keep on
going out we see we’ve got these gliders
in kind of outer space. And what’s interesting is
that now you look at it, you’re thinking it’s like these
little ants walking around. The grid, every cell, is only
responding to the eight cells around it. So there is no kind of
overall guiding thing, but yet there is. So Game of Life is
such a great way to understand how things
at different scales take on different appearances. So we keep on going
out of this pattern. All these are
gliders in streams. And here we are. In fact, what this
is– it’s a gun. It goes back in in
a second, magnifies. These are the two
kind of barrels. And everything is working
in such synchronicity that– going in. This is a spaceship, a special
slightly larger spaceship than a glider. That will be shot out. So this thing will keep
on going forever, just shooting out those spaceships. Now, what kind of things can you
build using the Game of Life? And I wanted to show
you this pattern. So this is a pattern,
20,000 cells. It’s the sieve of Eratosthenes. What it’s doing– it’s a gun
that’s shooting out spaceships only in the prime
number positions, OK? This is two, three. Four’s not prime
so it’s not there. That’s Five. Six is not there. That’s seven. That’s nine. Nine’s not going to come back
because a glider gets shot at it. OK. So what is actually
happening here is that, as that
goes along there, this thing here shoots
a glider every three. In other words, anything
that’s divisible by three it’s going to kill. This shoots something every
five, this every seven, this every nine. And if I go to the
next, this one here. Hang on. Yeah. So again. So this shooting every three. That’s seven. This is nine. Because we know all even
numbers aren’t prime, we only shoot at the odds. That is nine. That’s 11. That’s going to get through. So it’s the sieve
of Eratosthenes turned into this kind of
intergalactic shootout between gliders and spaceships. That’s 11. The next one is 15 so 15
is divisible by that’s the three coming in. That’s the five coming in. That’s 15. That’s going to get shot. So we could let this
pattern go forever and it would just
shoot out gliders. I’ve one slide to go left,
which is this one here. So what is the extent
of the Game of Life? What can the Game of Life do? It can find prime numbers. Actually, speeding up so we have
enough time, the Game of Life can do absolutely anything
that any computer can do. Right? Because what is a computer
at its most basic level? It’s a bunch of wires with
a pulse going through it. It’s binary. You can emulate a wire with
a pulse going through it by a stream of
gliders like here. So when there’s a
space it’s a zero. When there’s a
glider, there’s a one. What else do you need? You need logic gates. You can make the
three logic gates. There are patterns for
the three logic gates. And you need wires to be
able to cross each other. You can do that just by
thinning out the wires so they don’t hit. And you need a memory register. You can do that with a
block, which is the square. And a block at a certain
distance, keeping the memory. So the mathematical
term for that would be that Life is
universal because it’s capable of universal
computation. So even though it would be a
very inefficient way to do it, you could actually
use the Game of Life, which incredibly simple
rules can do anything that the most complicated,
sophisticated computer could ever do. And I guess on that note,
which is kind of quite a wow. We come to the end of the talk. That’s the name
of the book, which has more information both about
emotional responses to numbers, loads of things in
between, culminating in cellular automata. And the whole idea
about cellular automata is that it’s really good
for emulating replication. And people have now
started to get patterns that do self-replicate in
the sort of Von Neumann sense of self-replicating. And mathematicians there’s
only a matter of time that you will be able to create
self-replicating creatures that live on a kind of big infinite
grid of the Game of Life. So the kind of game
is turning into life. And there are philosophers who
say that, actually, the best explanation for the
world and the universe is a bit like the Game
of Life– that there’s some initial
configuration of kind of discrete elements of some way
and that we are just squillions of generations beyond
that beginning grid. So, again, it’s interesting
to think philosophically about that. That’s the first
book, “Numberland.” That’s the football
book that I did. I blog, “Alex’s Adventures in
Numberland,” at the “Guardian” every couple of weeks. And I’m on social
media if you want to follow any of the maths
popularization work that I do. Thank you very much. Sorry for talking for
a bit longer than I probably expected to. Thank you. PETER: Thank you so much. That was crazy stuff. Brilliant. And people have to
run, I’m afraid. It’s not rudeness. I think people have
meeting at 2:00. But I think we’ve probably
run out of time for questions. Except you neglected to say
what your favorite number was. ALEX BELLOS: Well,
that’s the last thing. I don’t have a
favorite number which is why I got interested
in it in the first place. I don’t have one. PETER: He’ll be around
for a few minutes. ALEX BELLOS: I will
totally be around. PETER: If you want to
ask questions one-to-one, feel free to do so. Thank you very much indeed. [APPLAUSE]

About the Author: Garret Beatty

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